RO-NMO 2010 Problem 121

Let $f:\mathbb{R}\to\mathbb{R}$ be a monotonic function and $F:\mathbb{R}\to\mathbb{R}$ given by $$F(x)=\int_0^xf(t)\ \text{d}t.$$ Prove that if $F$ has a finite derivative, then $f$ is continuous.

Dorin Andrica & Mihai Piticari